Strongly correlated quantum many-body systems exhibit fascinating phenomena such as the high-Tc superconductivity in cuprates, spin liquids in frustrated antiferromagnets and the fractional quantum Hall effects in 2D electron gas. However simulating the Hubbard model, which is the simplest model that captures the strong-correlation physics in high-Tc cuprates, turns out to be extremely difficult. In this talk, I will introduce tensor network states (TNSs) as the variational ansatze for simulating the ground state of the strongly correlated quantum many-body systems. The advantage of this method is that there is a solid theoretical justification developed using quantum information theory.
I will start by bridging the TNSs with the previously well-known variational wave-functions. Making such a connection allows us to embed previous knowledge in the TNSs framework, and benefit from the computational advantage of the TNSs method. To demonstrate the power of the TNSs method, I will focus on the frustrated spin-1/2 J_1-J_2 Antiferromagnetic Heisenberg model on a square lattice, approaching the ground state from a designed tensor structure and through a black-box search [1,2].
Its advantage over the Density Matrix Renormalization Group method (DMRG) will be shown. In the last part, I will discuss the recent developments with TNSs. Examples include examining boundary excitations through the entanglement spectra of the reduced density matrix; and probing the bulk excitations via the eigenvalue spectra of the transfer matrix of the TNSs . These recent developments make TNSs the perfect tool for studying the topologically ordered phase (such as spin liquids) as well as the conventional symmetry-breaking phase of matter.
 L. Wang, D. Poilblanc, Z.-C. Gu, X.-G. Wen and F. Verstraete, Phys. Rev. Let. 111, 037202 (2013).
 L. Wang, Z.-C. Gu, F. Verstraete and X.-G. Wen, arXiv:1112.3331.
 L. Wang, A. Essin, M. Hermele and O. Motrunich, arXiv:1409.7013.